462PolsWithMultiplZerosAndSolvDynSyst181216Rev Polynomials with Multiple Zeros and Solvable Dynamical Systems including Models in the Plane with Polynomial Interactions Francesco Calogeroa,b,1 and Farrin Payandeha,c,2 a Physics Department, University of Rome ”La Sapienza”, Rome, Italy b INFN, Sezione di Roma 1 c Department of Physics, Payame Noor University (PNU), PO BOX 19395-3697 Tehran, Iran 1 [email protected], [email protected] 2 f [email protected], [email protected] Abstract The interplay among the time-evolution of the coefficients ym (t) and the zeros xn (t) of a generic time-dependent (monic) polynomial provides a conve- nient tool to identify certain classes of solvable dynamical systems. Recently this tool has been extended to the case of nongeneric polynomials characterized by the presence, for all time, of a single double zero; and subsequently signifi- cant progress has been made to extend this finding to the case of polynomials featuring a single zero of arbitrary multiplicity. In this paper we introduce an approach suitable to deal with the most general case, i. e. that of a nongeneric time-dependent polynomial with an arbitrary number of zeros each of which features, for all time, an arbitrary (time-independent) multiplicity. We then focus on the special case of a polynomial of degree 4 featuring only 2 different zeros and, by using a recently introduced additional twist of this approach, we thereby identify many new classes of solvable dynamical systems of the following type: (n) arXiv:1904.00496v1 [math-ph] 31 Mar 2019 x˙ n = P (x1, x2) , n =1, 2 , (n) with P (x1, x2) two polynomials in the two variables x1 (t) and x2 (t). 1 Introduction Notation 1.1. Hereafter t generally denotes time (the real independent vari- able); (partial) derivatives with respect to time are denoted by a superimposed dot, or in some case by appending as a subscript the independent variable t preceded by a comma; all dependent variables such as x, y, z (often equipped 1 with subscripts) are generally assumed to be complex numbers, unless otherwise indicated (it shall generally be clear from the context which of these and other quantities depend on the time t, as occasionally—but not always—explicitly indicated); parameters such as a, b, c, α, β, γ, A, etc. (often equipped with subscripts) are generally time-independent complex numbers; and indices such as n, m, j, ℓ are generally positive integers, with ranges explicitly indicated or clear from the context. Some time ago the idea has been exploited to identify dynamical systems which can be solved by using as a tool the relations between the time evolutions of the coefficients and the zeros of a generic time-dependent polynomial [1]. The basic idea of this approach is to relate the time-evolution of the N zeros xn (t) of a generic time-dependent polynomial pN (z; t) of degree N in its argument z, N N N N−m pN (z; t)= z + ym (t) z = [z − xn (t)] , (1) m=1 n=1 X Y to the time-evolution of its N coefficients ym (t). Indeed, if the time evolution of the N coefficients ym (t) is determined by a system of Ordinary Differential Equations (ODEs) which is itself solvable, then the corresponding time-evolution of the N zeros xn (t) is also solvable, via the following 3 steps: (i) given the initial values xn (0) , the corresponding initial values ym (0) can be obtained from the explicit formulas expressing the coefficients ym of a polynomial in terms of its zeros xn reading (for all time, hence in particular at t = 0) N M m ym (t) = (−1) [xnℓ (t)] , m =1, 2, ..., N ; (2) ≤ 1 2 m≤ ( ) 1 n <n X<...<n N ℓY=1 (ii) from the N values ym (0) thereby obtained, the N values ym (t) are then evaluated via the—assumedly solvable—system of ODEs satisfied by the N co- efficients ym (t); (iii) the N values xn (t)—i. e., the N solutions of the dynamical system satisfied by the N variables xn (t)—are then determined as the N zeros of the polynomial, see (1), itself known at time t in terms of its N coefficients ym (t) (the computation of the zeros of a known polynomial being an algebraic operation; of course generally explicitly performable only for polynomials of degree N ≤ 4). Remark 1-1. In this paper the term ”solvable” generally characterizes sys- tems of ODEs the initial-values of which are ”solvable by algebraic operations”— possibly including quadratures yielding implicit solutions, generally also requir- ing the evaluation of parameters via algebraic operations. And let us emphasize that, because of the algebraic but nonlinear character of the relations between the zeros and the coefficients of a polynomial, it is clear that, even to relatively trivial evolutions of the N coefficients ym (t) of a time-dependent polynomial, there correspond much less trivial evolutions of its N zeros xn (t). On the other hand the fact that a time evolution is algebraically solvable has important im- plications, generally excluding that it can be ”chaotic”, indeed in some cases allowing to infer important qualitative features of the time evolution, such as 2 the property to be isochronous or asymptotically isochronous (see for instance [2] [3] [4]). The viability of this technique to identify solvable dynamical systems de- pends of course on the availability of an explicit method to relate the time- evolution of the N zeros of a polynomial to the corresponding time-evolution of its N coefficients. Such a method was indeed provided in [1], opening the way to the identification of a vast class of algebraically solvable dynamical systems (see also [5] and references therein); but that approach was essentially restricted to the consideration of linear time evolutions of the coefficients ym (t). A development allowing to lift this quite strong restriction emerged relatively recently [6], by noticing the validity of the identity −1 N N x˙ = − (x − x ) y˙ (x )N−m (3) n n ℓ m n 6 m=1 ℓ=1Y, ℓ=n X h i which provides a convenient explicit relationship among the time evolutions of the N zeros xn (t) and the N coefficients ym (t) of the generic polynomial (1). This allowed a major enlargement of the class of algebraically solvable dynamical systems identifiable via this approach: for many examples see [7] and references therein. Remark 1-2. Analogous identities to (3) have been identified for higher time-derivatives [8] [9] [7]; but in this paper we restrict our treatment to dy- namical systems characterized by first-order ODEs, postponing the treatment of dynamical systems characterized by higher-order ODEs (see Section 6). A new twist of this approach was then provided by its extension to non- generic polynomials featuring—for all time—multiple zeros. The first step in this direction focussed on time-dependent polynomials featuring for all time a single double zero [10]; and subsequently significant progress has been made to treat the case of polynomials featuring a single zero of arbitrary multiplic- ity [11]. In Section 2 of the present paper a convenient method is provided which is suitable to treat the most general case of polynomials featuring an ar- bitrary number of zeros each of which features an arbitrary multiplicity. While all these developments might appear to mimic scholastic exercises analogous to the discussion among medieval scholars of how many angels might dance simul- taneously on the point of a needle, they do indeed provide new tools to identify new dynamical systems featuring interesting time evolutions (including systems displaying remarkable behaviors such as isochrony or asymptotic isochrony: see for instance [10] [11]); dynamical systems which—besides their intrinsic mathe- matical interest—are quite likely to play significant roles in applicative contexts. Such developments shall be reported in future publications. In the present paper we focus on another twist of this approach to identify new solvable dy- namical systems which was introduced quite recently [12]. It is again based on the relations among the time-evolution of the coefficients and the zeros of time- dependent polynomials [6] [7] with multiple roots (see [10], [11] and above); but (as in [12]) by restricting attention to such polynomials featuring only 2 zeros. 3 Again, this might seem such a strong limitation to justify the doubt that the results thereby obtained be of much interest. But the effect of this restriction is to open the possibility to identify algebraically solvable dynamical models characterized by the following system of 2 ODEs, (n) x˙ n = P (x1, x2) , n =1, 2 , (4) (n) with P (x1, x2) two polynomials in the two dependent variables x1 (t) and x2 (t); hence systems of considerable interest, both from a theoretical and an applicative point of view (see [12] and references quoted there). This develop- ment is detailed in the following Section 3 by treating a specific example. In Section 4 we report—without detailing their derivation, which is rather obvious on the basis of the treatment provided in Section 3—many other such solvable models (see (4); but in some cases the right-hand side of these equations are not quite polynomial); and a simple technique allowing additional extensions of these models—making them potentially more useful in applicative contexts—is outlined in Section 5, by detailing its applicability in a particularly interesting case. Hence researchers primarily interested in applications of such systems of ODEs might wish to take first of all a quick look at these 2 sections. Finally, Section 6 outlines future developments of this research line; and some material useful for the treatment provided in the body of this paper is reported in 2 Appendices.